Edge perturbations on signed graphs with clusters
Let $\Gamma$ be a signed graph. A cluster in $\Gamma$ of order $c$ and degree $s$, is a pair of vertex subset $(C,S)$, where $C$ is a set of cardinality $c \geq 2$ of pairwise co-neighbor vertices sharing the same set of $s$ neighbors and all edges connecting a fixed vertex in $C$ are equallly signed. We consider the graph $\Gamma(H)$ which is obtained from $G$ by identifying $V(H)$ with $C$ and show that some Laplacian or Adjacency eigenvalues of $\Gamma(H)$ remain the same whatever $H$ we choose in a suitable set of signed graphs.
Such techniques also provide a generalization to signed contexts of the Faria’s lower bound on the multiplicity of the Laplacian eigenvalue 1 of a graph with pendant vertices.